Theorem iscmgmALT 44125

Description: The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ismgmALT.b	⊢ 𝐵 = (Base‘𝑀)
ismgmALT.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
iscmgmALT	⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵))

Proof of Theorem iscmgmALT

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6664	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		fveq2 6664	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3	1, 2	breq12d 5071	. . 3 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ (+g‘𝑀) comLaw (Base‘𝑀)))
4		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	breq12i 5067	. . 3 ⊢ ( ⚬ comLaw 𝐵 ↔ (+g‘𝑀) comLaw (Base‘𝑀))
7	3, 6	syl6bbr 291	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ ⚬ comLaw 𝐵))
8		df-cmgm2 44121	. 2 ⊢ CMgmALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
9	7, 8	elrab2 3682	1 ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   class class class wbr 5058  ‘cfv 6349  Basecbs 16477  +_gcplusg 16559   comLaw ccomlaw 44086  MgmALTcmgm2 44116  CMgmALTccmgm2 44117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-iota 6308  df-fv 6357  df-cmgm2 44121

This theorem is referenced by: (None)